John Allen Paulos – ABCNEWS.com

One generally doesn’t speak the words “prime numbers” and “seven-figure prizes” in the same breath.
But don’t tell that to the publishers of “Uncle Petros and Goldbach’s Conjecture”, an engaging first novel by Greek author Apostolos Doxiadis. The Story Behind the Math

Before getting to the money, here’s a quick synopsis of the story: The narrator tells of his Uncle Petros, whom he initially thinks of as the eccentric black sheep of the family.

Slowly, Uncle Petros is revealed to be a character of complexity and nuance, having devoted his considerable mathematical talents and much of his life to a futile effort to prove a classic unsolved problem. His solitary efforts give one a taste of the delight and the despair of mathematical research. Goldbach’s Conjecture, Uncle Petros’ holy grail, is startlingly simple to state: Any even number greater than 2 is the sum of two prime numbers. Remember that a prime number is a positive whole number that is divisible only by two numbers: itself and 1; thus 5 is a prime, but 6, which is divisible by 2 and 3, is not. The number 1 is not considered prime. Check out the claim. Pick an even number at random and try to find two primes which add up to it. Certainly, 6 = 3 + 3, 20 = 13 + 7, and 97 + 23 =120. (This, of course, is not a proof.) The conjecture that this works for every even number greater than 2 was proposed in 1742 by Prussian mathematician Christian Goldbach. To this day it remains unproved despite the efforts of some of the world’s best mathematicians. The Frustrations of Whole Numbers

Number theory, the branch of mathematics that studies prime numbers and other ethereal aspects of the integers (whole numbers), contains many problems that are easy to state and yet resistant, so far, to the efforts of all. The Twin Primes Conjecture is another: There are an infinite number of prime pairs, prime numbers that differ by 2.Examples are 5 and 7, 11 and 13, 17 and 19, 29 and 31, and, presumably, infinitely many more. Of more contemporary origin is the so-called Collatz Conjecture, sometimes called the 3x + 1 problem. Choose any whole number. (Take 13, for example) If it is odd, multiply it by 3 and add 1. (3 times 13 plus 1 equals 40.) If it is even, divide it by 2. (40 divided by 2 is 20.) Continue this procedure with each resulting number and the conjecture is that the sequence thus generated always ends up 4, 2, 1, 4, 2, 1, 4, … The sequence starting with 13, produces 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, …. Every number that’s been tried (up to about 27 quadrillion) ultimately cycles back to 4, 2, 1, but there is still no proof that every number does. Like the recently proved Fermat’s Last Theorem (xn + yn = zn, where x, y, z, n are all integers, has no solutions for n > 2), these conjectures are tantalizing and can sometimes become, if one is not careful, all-consuming obsessions. Such an obsession is the fate of Uncle Petros. His brothers think little of him and of his quixotic attempt to prove Goldbach’s Conjecture. Petros in turn has disdain for their petty concern with the family business in Athens. Interestingly, Petros also has a low regard for applied mathematics, which he compares to glorified “grocery bill” calculations and which, he believes, shares none of the austere beauty of pure number theory. Theory vs. Practical
This mutual contempt between mathematicians and more practical sorts has a long history. The British mathematician G.H. Hardy, a colleague of the fictional Uncle Petros in the book, exulted in the uselessness of mathematics, particularly number theory. Happily, this adversarial attitude has softened in recent years, and even number theory, arguably the most impractical area of math, has found important applications. Cryptographic codes, which enable the transfer of trillions of dollars between banks, businesses, and governments, depend critically on number theory. They depend, in particular, on the simple fact that multiplying two large prime numbers together is easy, but factoring a large number (say one having 100 digits) into prime factors is extraordinarily difficult and time-consuming. Finally, I come to the million dollar contest. The U.S. publisher of Uncle Petros and Goldbach’s Conjecture has promised $1 million to the first person to prove the conjecture, provided the proof appears in a reputable mathematics journal before 2004. The late, great number theorist Paul Erdos used to offer small monetary prizes to anyone solving this or that problem, but he didn’t have to pay up often. If I were the publisher, I wouldn’t worry about the offer’s financial risk, but I would be apprehensive about the torrent of false proofs that will soon be heading their way.

Professor of mathematics at Temple University, John Allen Paulos is the author of several books, including A Mathematician Reads the Newspaper and, most recently, I Think, Therefore I Laugh. His “Who’s Counting?” column on ABCNEWS.com appears on the first day of every month.

Solution to the Small Challenge

From the integers between 1 and 100, choose any subset of 10 numbers. Call it S. How many subsets does S have? A preliminary observation is that a set with 2 numbers in it — say a and b — has 22 – 1, or 3, subsets: {a}, {b}, and {a, b}. A set with 3 numbers in it — say a, b, and c — has 23 – 1, or 7, subsets: {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, and {a, b, c}. More generally, one can show that a set with n numbers in it has 2n – 1 non-empty subsets. Since the set S contains 10 numbers, it has 210 – 1, or 1,023, subsets. But how many possible sums are there for the numbers in each of these 1,023 subsets of S? Even if for S you chose the 10 largest numbers, 91, 92, 93,…100, the sum is less than 1,000 (955, to be exact). Thus, for any subset of S, the sum would certainly be less than 1,000. That, in turn, means whatever S you choose, there are fewer than 1,000 possible sums for the numbers in each of its subsets. Since 1,023 is greater than 1,000, there are always more subsets of S than there are possible sums for the numbers in each of the subsets. Thus at least two of the subsets of S must have the same sum.